In this paper, we show that the Kirchhoff equations are derived from the Schrödinger equation by assuming the wave function to be a polynomial like solution. These Kirchhoff equations describe the evolution of
n point vortices in hydrodynamics. In two dimensions, Kirchhoff equations are used to demonstrate the solution to single particle Laughlin wave function as complex Hermite polynomials. We also show that the equation for optical vortices, a two dimentional system, is derived from Kirchhoff equation by using paraxial wave approximation. These Kirchhoff equations satisfy a Poisson bracket relationship in phase space which is identical to the Heisenberg uncertainty relationship. Therefore, we conclude that being classical equations, the Kirchhoff equations, describe both a particle and a wave nature of single particle quantum mechanics in two dimensions.
In this paper, we show that in two dimensions quantum mechanics can be mapped onto classical mechanics, by transforming the Schr\odinger equation into system of
n linear equations known as Kirchhoff equations. These Kirchhoff equations equations satisfy the a poison bracket relationship in phase space which is identical to the Heisenberg uncertainty relationship. Therefore, we conclude that quantum mechanics is consistent with classical mechanics atleast in two dimensions. This allows us to address the wave particle duality in terms of relative phase. As an illustration we show that the equation for optical vortices can be derived as Kirchhoff equation admit a paraxial wave equation in presence of real constant background.